Project: Hubble Diagrams Distances Exercise 1 In this exercise, you will find the magnitudes of six galaxies in the SDSS database. The table below shows the object IDs and positions (right ascension and declination) of the six galaxies. Question 1: Why can magnitudes be used as a substitute for distances in the Hubble diagram? Magnitudes can be used as a substitute for distances because magnitude varies with distance. Redshifts Exercise 2 Find redshifts for the galaxies that you used in Exercise 1. Making the Diagram Exercise 3 Follow the steps below to make a simple Hubble diagram for six galaxies. Exercise 4 Find the fit of a linear model in your Hubble diagram. Another Hubble Diagram Exercise 5 Repeat Exercises 1 and 2 for the following galaxies: Exercise 6 Repeat Exercise 3 for these six galaxies. Graph these data on the same scale you used in Exercise 3. What do your data look like now? Repeat Exercise 4. What is the percentage fit of the data? Estimating Distances to Galaxies Radiant flux: F = 2.51 -m F is a relative number that compares the arriving radiant flux to the star Vega.
Relative distance to a galaxy is the inverse square root of F. Normalize by solving d 1 / d 2 = 1 / x for x for each galaxy, where d 1 is the relative distance to the nearest galaxy and d 2 is the relative distance to another galaxy and x is the normalized distance to the other galaxy. Relative distance can also be estimated by comparing apparent sizes of galaxies. Measure the width of the image of each galaxy. Compare the inverse of this measure to get relative distance. This approach assumes that galaxies have about the same true size. Exercise 7 Find the relative distances between the six galaxies whose magnitudes you found in Exercise 1. Use a scientific calculator that can display numbers in scientific notation (that is, as 1.5 million = 1.5E+06). Exercise 8 Write the two techniques for finding relative distance as algebraic equations. Derive them using geometrical or physical principles. Relative distance... from radiant flux: d = 1 2.51 m, where d is relative distance and m is apparent magnitude from apparent size: d = 1 w, where w is the apparent width of a galaxy. Question 2: Suppose the relative distances for a number of galaxies using brightnesses don't agree with the relative distances using apparent sizes. What would you conclude? I would conclude that not all galaxies are the same true size. Estimating Distances to Clusters Question 3: Why can we assert that galaxies in a cluster are all at the same distance? I think we can make this assertion because clusters can be thought of as statistical units or populations of galaxies. Statistically, they are at the same distance. Exercise 9 Show that the fractional error in the assumption that galaxies in a spherical cluster are all at the same distance is equal to the cluster's angular size: the angle of the sky that it takes up when viewed from Earth.
Assuming the width and depth of a spherical cluster to be equal, the largest error in distance will be the depth of the cluster which is also its width, its angular size. Question 4: What are some of those clues and cues? Would any of those techniques apply to estimating relative distances for galaxies in space? In surveying the countryside for towns, cities, and buildings, clues abound that would help distinguish one from the next. Building types, knowledge of relative building sizes, separation distance between buildings, etc. Some of those techniques might be helpful in estimating relative distances for galaxies in space, especially accepted separation distance. Exercise 10 Look at SDSS images for the following clusters: table follows. For each cluster, think about how we know that the galaxies are actually part of the same cluster. What properties are similar between galaxies in the same cluster? What properties show a wide range? How might you be able to tell - using just these images - if any particular galaxy is actually in the cluster, as opposed to being at a different distance along the same line-of-sight? Similar properties: color, magnitude Wide range: size Question 5: What would tell Hubble and Humason that one approach was better than another? Assuming that the brightest galaxy in one cluster should have about the same true brightness as the brightest galaxy in another cluster, approaches that result in closer brightness values might be deemed better. Relative Distances for Sample Galaxies Exercise 11 Look at the SDSS image at right. The image shows three galaxy clusters in the same area of the sky. Look closely at the image and decide which galaxies belong to which clusters. Make some notes for yourself about which galaxies belong where. I decided to invert the image so that objects would more easily be seen. I think the three galaxies are in the areas circled.
Exercise 12 Now, find the relative distances to the galaxies you studied in Exercise 11. Exercise 13 Repeat Exercise 12 for the same clusters using a different measured quantity leading to another estimate of relative distance. Add two columns to the right edge of your table for your second measurement and second relative distance. How do your independent estimates of the cluster distances compare? Which is better? Why? Rather than using additional columns in one table, a table was created for each additional measurement. Apparent magnitudes of five wavelengths were used as the measurements. Four of the five put the same galaxy as the closest. The two infrared wavelengths agree in the rank order among the galaxies. Red and green wavelengths are somewhat similar to each other while ultraviolet is quite different than all other measurements. I do not think any one estimate is inherently better than the others because stars emit in all these wavelengths differently. Perhaps some averaging method should be employed. Redshifts Measuring a redshift or blueshift requires four steps: 1) obtain the spectrum of something (let's say a galaxy) that shows spectral lines 2) from the pattern of lines, identify which line corresponds to which atom, ion, or molecule 3) measure the shift of any one of those lines with respect to its expected wavelength, as measured in a laboratory on Earth 4) apply a formula that relates the observed shift to velocity along the line-of-sight Redshift is symbolized as 1 z = l observed l rest so that z= l observed l rest 1 where z is the redshift (negative indicates blueshift), l observed and l rest are individual Balmer lines for a particular wavelength observed and at rest, respectively. Interpreting Redshifts Speed of a galaxy toward/away from us is v = c z where c is the speed of light (3.00 10 5 km/sec), z is the redshift, and v is the speed in km/sec. Question 6: In the SkyServer database, you can find redshifts for quasars such that z > 1. Is there a conceptual problem if the redshift is interpreted as a Doppler shift velocity? Is there a conceptual problem if the redshift is interpreted as the cosmological stretching of space? There is a conceptual problem if the redshift is interpreted as a Doppler shift velocity because the velocity would then be greater than the speed of light. There is also a conceptual problem if the redshift is interpreted as the cosmological stretching of space because z = 1 would correspond to a time when galaxies were 100% closer together than they are now.
Exercise 15 Redshift templates. The application did not work. Exercise 16 Open your online notebook containing the galaxies you found relative distances to in the last section. Of the objects you selected, at least four should have spectra available in the SDSS spectra database. In fact, these four were among the ten galaxies you found redshifts for in the last exercise. The table below tells you which spectrum number from Exercise 13 corresponds to which galaxy's object ID from the Distances section. Click any of the object IDs to open the Object Explorer in the tools window. Write down the redshift (the "z" just above the spectrum) for each galaxy. Spectrum Number Galaxy ID 527 587722984423686445 530 587722984423686301 523 587722984423686312 525 587722984423686395 Compare the redshifts found by the SDSS with the redshifts you calculated in Exercise 15. How close were you? The application to calculate redshift did not work. Exercise 17